Infrastructure for location discovery

ABSTRACT

Techniques are generally described for determining locations of a plurality of communication devices in a network. In some examples, methods for creating a location discovery infrastructure (LDI) for estimating locations of one or more of a plurality of communication nodes may comprise one or more of determining a plurality of locations in the terrain to place a corresponding plurality of beacon nodes, determining a plurality of beacon node groups for the placed beacon nodes, and determining a schedule for the placed beacon nodes to be active. Additional variants and embodiments are also disclosed.

RELATED APPLICATIONS

This application is a continuation-in-part of co-pending applicationSer. No. 12/415,518, filed on Mar. 31, 2009, entitled “EfficientLocation Discovery,” and co-pending application Ser. No. 12/415,523,filed on Mar. 31, 2009, entitled “Network Node Location Discovery,” theentire specifications of which are hereby incorporated by reference inits entirety for all purposes, except for those sections, if any, thatare inconsistent with this specification.

BACKGROUND

Wireless communication networks are becoming increasingly popular. Awireless network may include plurality of wireless devices. In someapplications, the locations of some of the wireless devices may beknown, while the location of one or more remaining wireless devices inthe wireless network may need to be determined. Such determination maybe useful for a variety of applications, such as navigation, tracking,and so forth. Improved determination techniques may facilitate theeffectiveness of many sensing and communication procedures.

BRIEF DESCRIPTION OF THE DRAWINGS

Subject matter is particularly pointed out and distinctly claimed in theconcluding portion of the specification. The foregoing and otherfeatures of this disclosure will become more fully apparent from thefollowing description and appended claims, taken in conjunction with theaccompanying drawings. Understanding that these drawings depict onlyseveral embodiments in accordance with the disclosure and are,therefore, not to be considered limiting of its scope, the disclosurewill be described with additional specificity and detail through use ofthe accompanying drawings. Various embodiments will be describedreferencing the accompanying drawings in which like references denotesimilar elements, and in which:

FIG. 1 illustrates an example wireless network;

FIG. 2 illustrates an example terrain that includes one or moreobstacles;

FIG. 3 illustrates an example method for creating and operating aninfrastructure for determining locations of one or more communicationnodes;

FIG. 4 illustrates an example method for constructing a distance errormeasurement model;

FIG. 5 illustrates an example distance measurement model;

FIG. 6 illustrates another example distance measurement model;

FIG. 7 illustrates an example outdoor terrain model;

FIG. 8 illustrates an example indoor terrain model;

FIG. 9 a illustrates an example model that includes correlation betweena number of neighboring nodes of individual communication nodes and acorresponding cumulative density function (CDF) values;

FIG. 9 b illustrates an example location discovery error model thatincludes correlation between the CDF values of number of neighboringnodes of individual communication nodes and the corresponding locationestimation error;

FIG. 10 a illustrates an example wireless network;

FIG. 10 b illustrates an example model that includes correlation betweenthe third largest angle of neighboring nodes of individual communicationnodes and the corresponding CDF values;

FIG. 10 c illustrates an example location discovery error model thatincludes correlation between the CDF values of the third largest angleof neighboring nodes of individual communication nodes and thecorresponding location estimation error;

FIG. 11 illustrates an example location discovery error model thatincludes correlation between the third largest angle of neighboringnodes of individual communication nodes, the number of neighboring nodesof corresponding communication nodes, and the location estimation errorof the corresponding communication nodes;

FIG. 12 illustrates an example method for determining a plurality oflocations in a terrain to place a corresponding plurality of beaconnodes;

FIG. 13 illustrates an example method for determining a plurality ofbeacon node groups for the placed beacon nodes;

FIG. 14 illustrates an example method for scheduling signal transmissionof a plurality of beacon nodes;

FIG. 15 illustrates an example computing system that may be suitable forpracticing various embodiments; and

FIG. 16 illustrates an example computing program product, all arrangedin accordance with the present disclosure.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The following description sets forth various examples along withspecific details to provide a thorough understanding of claimed subjectmatter. It will be understood by those skilled in the art, however, thatclaimed subject matter may be practiced without some or more of thespecific details disclosed herein. Further, in some circumstances,well-known methods, procedures, systems, components and/or circuits havenot been described in detail in order to avoid unnecessarily obscuringclaimed subject matter. In the following detailed description, referenceis made to the accompanying drawings, which form a part hereof. In thedrawings, similar symbols typically identify similar components, unlesscontext dictates otherwise. The illustrative embodiments described inthe detailed description, drawings, and claims are not meant to belimiting. Other embodiments may be utilized, and other changes may bemade, without departing from the spirit or scope of the subject matterpresented here. It will be readily understood that the aspects of thepresent disclosure, as generally described herein, and illustrated inthe Figures, may be arranged, substituted, combined, and designed in awide variety of different configurations, all of which are explicitlycontemplated and make part of this disclosure.

In the following description, algorithms and/or symbolic representationsof operations on data bits and/or binary digital signals stored within acomputing system, such as within a computer and/or computing systemmemory may be presented. An algorithm is generally considered to be aself-consistent sequence of operations and/or similar processing leadingto a desired result where the operations may involve physicalmanipulations of physical quantities that may take the form ofelectrical, magnetic and/or electromagnetic signals capable of beingstored, transferred, combined, compared and/or otherwise manipulated. Invarious contexts such signals may be referred to as bits, data, values,elements, symbols, characters, terms, numbers, numerals, etc. Thoseskilled in the art will recognize, however, that such terms may be usedto connote physical quantities. Hence, when terms such as “storing”,“processing”, “retrieving”, “calculating”, “determining” etc. are usedin this description they may refer to the actions of a computingplatform, such as a computer or a similar electronic computing devicesuch as a cellular telephone, that manipulates and/or transforms datarepresented as physical quantities including electronic and/or magneticquantities within the computing platform's processors, memories,registers, etc.

This disclosure is drawn, inter alia, to methods, apparatus, systems andcomputer program products related to an infrastructure for locationdiscovery of one or more communication nodes in a wireless network.

FIG. 1 illustrates an example wireless network 10, in accordance withvarious embodiments of the present disclosure. The network 10 may be anyappropriate type of wireless network, including but not limited to awireless ad hoc network, a mobile ad hoc network, a wireless meshnetwork, a wireless sensor network, or the like. The wireless network 10may include a plurality of communication nodes Sa, . . . , Sf and aplurality of beacon nodes Ba, . . . , Bd. In various embodiments,individual communication nodes Sa, . . . , Sf and/or the beacon nodesBa, . . . , Bd may be capable of communicating with other nodes and/orother wireless devices within network 10 using an appropriate wirelessprotocol, such as but is not limited to Institute of Electrical andElectronics Engineers (IEEE) 802.11 standards (e.g., 802.11g. releasedon June, 2003). In various embodiments, one or more communication nodesmay comprise of one or more appropriate wireless communication devices.

As illustrated in FIG. 1, an X-axis and a Y-axis may be analyticallysuperimposed on the network 10 to permit locating one or more nodes withrespect to the two axes. The number and location of the nodesillustrated in FIG. 1 may be purely an example. In various otherembodiments, the disclosure may be practiced with more or less nodes ofeither type or types, or different locations of the nodes. For example,although the nodes ma y be illustrated to be located in the firstquadrant of the coordinate system of FIG. 1 (e.g., positive x and ycoordinates), in various embodiments, one or more beacon nodes and/orcommunication nodes may also be located in other quadrants as well.

In various embodiments, locations of one or more beacon nodes Ba, . . ., Bd may be known relative to the two analytical axes (hereinafter,simply axes). For example, the location coordinates of the beacon nodeBa (e.g., (X_(Ba), Y_(Ba))) with reference to the X-axis and the Y-axismay be known to the beacon node Ba. Similarly, the coordinates of thebeacon nodes Bb, . . . , Bd (e.g., (X_(Bb), Y_(Bb)), . . . , (X_(Bd),Y_(Bd))) may also be known to the respective beacon nodes. In variousembodiments, one or more beacon nodes may be equipped with a globalpositioning system (GPS) or any other appropriate locationidentification system through which the beacon nodes may identify theirrespective locations. In other embodiments, some or all the beacon nodelocations may be predetermined. In various embodiments, the locationinformation may be shared with one or more peer or server devices.

In various applications, it may not be feasible to predetermine or equipall the nodes in the network with GPS or other location identificationsystem due to, for example, high cost, low battery life, larger size,weight, etc. of such a system. Accordingly, in various embodiments,location of one or more communication nodes Sa, . . . , Sf may not bepre-known, and/or one or more communication nodes may not be equippedwith, for example, a GPS system. For example, the location coordinatesof the communication node Sa (e.g., (X_(Sa), Y_(Sa))) relative to theX-axis and the Y-axis may be unknown to the node Sa and/or to othernodes in the network 10. Thus, in various embodiments, the location(s)of one or more communication nodes in network 10 may be needed to bedetermined.

In various embodiments, there may be different types of communicationbetween two nodes of the network 10. For example, a node may be arrangedto transmit acoustic signals that may include information related tosuch a node (e.g., location information of such a node, if known), andmay also be arranged to receive acoustic signals transmitted by othernodes. In various embodiments, a node may also be arranged to transmitand receive radio signals including data and/or other information. Invarious embodiments, an acoustic signal range (ASR) of a node may beindependent from a radio signal range (RSR) of the node. Furthermore,all nodes in the network may not have the same ASR and RSR properties.

In various embodiments, reception of appropriate signals by a first nodefrom a second node may permit the first node to determine a distancebetween the first node and the second node. In various embodiments,determination of distance between two nodes may be performed usingtechniques known to those skilled in the art.

For example, a node may be configured to receive acoustic signals fromone or more neighboring nodes, and may also be configured to determinedistances between such a node and the one or more neighboring nodes inresponse to the received acoustic signals. In various embodiments, anode may be configured to receive acoustic signals from one or morenodes that are within an ASR of the node, wherein the acoustic signalsmay include location information about the respective one or more nodes.In various embodiments, other types of signals (e.g., radio signals) mayalso include location information about a node, and it may be possibleto determine a distance between two nodes based at least in part on oneof the nodes receiving a radio signal from another node.

For example, communication node Sa may receive signals from neighboringbeacon nodes Ba, Bb, and Bc, and may be able to determine distancesd_(aa), d_(ab), and d_(ac) between the communication node Sa and beaconnodes Ba, Bb, and Bc, respectively. In various embodiments,communication node Sa may be arranged to receive signal transmissionsfrom beacon node Bd when node Sa is within a signal range of node Bd,and in that case, node Sa may be arranged to determine a distance d_(ad)(not illustrated in FIG. 1) between node Sa and beacon node Bd.Similarly, other communication nodes Sb, . . . , Sf may also be arrangedto determine distances between the respective communication node and oneor more neighboring beacon nodes. In various embodiments, a firstcommunication node may also be configured to determine a distancebetween the first communication node and a second communication node.For example, communication node Sc may be configured to estimate adistance D_(ce) between communication nodes Sc and Se.

Transmission of a signal by a node (e.g., a first node) may imply thatanother node (e.g., second node) may be arranged to receive the signaland determine its distance from the node (e.g., the first node) inresponse to the received signal.

In various embodiments the phrase “network” may indicate a wirelesscommunication network (such as network 10 of FIG. 1) that may include aplurality of communication nodes (one or more of whose locations mayneed to be determined) and a plurality of beacon nodes (one or more ofwhose locations may be known or already determined). In variousembodiments, the number and/or configuration of communication and/orbeacon nodes in such a “network” may not be restricted by the numberand/or configuration of communication and/or beacon nodes in the network10 of FIG. 1.

In various embodiments, neighboring nodes of a selected node may includeat least those other nodes from which the selected node may receivesignal transmission that may permit the selected node (or vicariouslyvia a peer/server device) to determine the distance from the selectednode to those other nodes. Thus, a selected node may calculate adistance to other nodes that are determined to be neighboring nodes ofthe selected node, based on signal transmissions received by theselected node from the other nodes. For example, in the network 10 ofFIG. 1, communication node Sb may be arranged to receive transmittedsignals from nodes Ba, Bb, Sa, and Sc, determine the distances to thesenodes, and designate these nodes as neighboring nodes of node Sb.Similarly, when Sb is unable to determine its distance from node Bd (asSb may not receive signal from Bd because of, for example, a largedistance between the two nodes), then Bd may not be designated as aneighboring node of node Sb.

Although the nodes in FIG. 1 are illustrated to be in a two dimensionalplane (e.g., x-y plane), in various embodiments, the inventiveprinciples discussed in this disclosure may be extended to a threedimensional space as well (by adding a third vertical dimension orz-axis). Although a Cartesian coordinate system (e.g., with X-Y axis)has been used in FIG. 1, any other appropriate coordinate systems (e.g.,a polar coordinate system, a cylindrical coordinate system, a circularcoordinate system, or the like) may also be used.

In various embodiments, it may be possible to determine the location ofa selected communication node based on determining the distance betweenthe selected communication node and at least three non-collinearneighboring beacon nodes, if the determination of location is done in atwo dimensional plane, as will be discussed in more details. If thedetermination of location is done in a three dimensional plane, distancebetween the selected communication node and at least four non-collinearneighboring beacon nodes may be used. Without losing any generality andmainly for the purpose of simplicity, the location determination problemdiscussed henceforth in this disclosure may be considered in a twodimensional plane, unless otherwise stated. As will be readilyunderstood by those skilled in the art based on the teachings providedin this disclosure, the inventive principles of various embodiments ofthis disclosure may be easily applied for a higher (e.g., three)dimensional location determination.

Accordingly, to effectively determine the locations of one or morecommunication nodes, individual communication nodes may have at leastthree non-collinear neighboring beacon nodes. In various embodiments,estimation of distance between two nodes and subsequent estimation oflocation of a communication node may be related to the environment(e.g., the terrain) in which the nodes may be deployed. For example,some or all the nodes may be deployed in an indoor terrain and/or anoutdoor terrain, with one or more obstacles (e.g., trees, buildings,walls, etc.) in the terrain that may hinder communication between twonodes. Accordingly, given a terrain with one or more obstacles, aplurality of beacon nodes may be placed in the terrain. Individualcommunication nodes may be arranged to communicate with at least threenon-collinear neighboring beacon nodes (from the plurality of beaconnodes placed in the terrain) to determine the location of the individualcommunication nodes.

Although individual nodes in the network 10 may have been identifiedeither as a communication node or a beacon node, in various embodiments,the difference between the two types of nodes, for the purpose of thisdisclosure, may be the location knowledge that may be available to thebeacon nodes, as will be discussed in more details herein later. Invarious embodiments, the beacon nodes may also act as a communicationnode, and vice versa, and the labeling of a node as a communication nodeand/or a beacon node is not intended to limit other functionalities ofthe node. In various embodiments, a communication node may also act as abeacon node. For example, once the location of a communication node Sais determined using one of the various methods described in more detailslater in this disclosure, the node Sa may subsequently act as a beaconnode while facilitating determination of the locations of othercommunication nodes (e.g., nodes Sb, Sc, etc.). In various embodiments,a beacon node may also act as a communication node in case the beaconnode loses its location determination capability due to, for example,loss of its GPS capabilities, suspicious readings on the GPS system,etc. For the purpose of this disclosure and unless otherwise stated, thephrase node (without specific mention of its type) may refer to acommunication node and/or a beacon node.

Overview of Infrastructure for Location Discovery (LDI)

FIG. 2 illustrates an example terrain 200 that includes one or moreobstacles, in accordance with various embodiments of the presentdisclosure. The shaded regions in FIG. 2 may illustrate exampleobstacles (e.g., obstacles 210, 214, etc.), which may be of any type(e.g., houses, buildings, trees, walls, etc.), and which may block allor part of acoustic signals used for estimation of locations ofcommunication nodes. The shape and size of the terrain 200 and theobstacles 210 and 214 is merely an example, and other shapes and sizesof terrain and obstacles may also be possible. In various embodiments,the terrain 200 may be 500 meters (m) by 500 m, with known topology(e.g., known shape and size of the obstacles), and may be divided in a5×5 grid 220, as illustrated by the solid lines.

In various embodiments, a minimal number of beacon nodes may be placedin the terrain 200 such that location of an arbitrary placedcommunication node in the terrain 200 may be determined. The placedbeacon nodes may be a part of an infrastructure for location discovery(LDI) of one or more communication nodes in the terrain. In variousembodiments, the terrain 200 may include a plurality of beacon nodes Ba,Bb, . . . , Bi, the placement of which will be discussed in more detailslater.

FIG. 3 illustrates an example method 300 for creating and operating aninfrastructure for determining locations of one or more communicationnodes in the terrain, in accordance with various embodiments of thepresent disclosure. Various ones of these embodiments may beinstantiated on a computing device, that is, practicing the method on acomputing device provided with the decision logic to be described. Themethod 300 may include, at block 304, determining a plurality oflocations in a terrain (e.g., terrain 200 of FIG. 2) to place acorresponding plurality of beacon nodes, based at least in part onlikelihood of location discovery error for discovering locations of aplurality of communication nodes using the placed beacon nodes. Forexample, the beacon nodes may be placed such that a communication nodelocated in any part of the terrain may have three or more neighboringbeacon nodes. The beacon nodes may also be placed in such a way that thelocation determination errors may be reduced for a set of stationary ormobile communication nodes. An example placement of nine beacon nodes(Ba, . . . , Bi) is illustrated in FIG. 2, where a communication nodelocated in any part of the terrain can communicate, without anyobstruction, with three or more neighboring beacon nodes, under theassumption that the communication range of the communication and/orbeacon nodes is at least 2.5 m. In various embodiments, a minimum numberof beacon nodes and their corresponding locations for placement in theterrain may also be determined, such that a communication node locatedanywhere in the terrain may receive signals from three or moreneighboring beacon nodes. The placement of the beacon nodes will bediscussed in more details herein later.

Referring again to FIG. 3, the method 300 may further include, at block308, determining a plurality of beacon node groups for the placed beaconnodes, based at least in part on the placed beacon nodes withinindividual beacon node groups being able to be active simultaneouslywithout causing interference or collision at the individualcommunication nodes of the plurality of communication nodes. Forexample, at block 308, the nine placed beacon nodes in FIG. 2 may bedivided in one or more beacon node groups, such that the individualbeacon nodes in a beacon node group may simultaneously broadcast itsacoustic signals simultaneously without causing collision at thereceiving communication nodes. For example, beacon nodes Ba and Bf ofFIG. 2 may simultaneously transmit signals without causing collision orinterference at any location in the terrain 200. Similarly, beacon nodesBc and Bg may also transmit simultaneously without causing collision orinterference at any location in the terrain. In various embodiments,beacon nodes Ba and Bb, for example, may be proximally located, andthere may be one or more communication nodes (not illustrated in FIG. 2)in the terrain that may receive signals from both the beacon nodes Baand Bb. Simultaneous transmission from beacon nodes Ba and Bb may causecollision or interference at such communication nodes. Accordingly,beacon nodes Ba and Bb may not transmit simultaneously.

In various embodiments, a possible example grouping of beacon nodes maybe {{Ba, Bf}, Bb, {Bc, Bg}, Bd, Be, Bh, Bi}. For this example, there isa total of seven groups: two groups having two beacon nodes each, andrest of the five groups having a single beacon node. Without thegrouping, nine separate beacon nodes would transmit in their ownrespective time slots, thereby requiring a total of nine time slots.However, with the described example grouping, only seven time slots maybe required for the nine beacon nodes, one slot for a respective beaconnode group. Another example grouping may be {{Ba, Bf}, {Bb, Bh}, {Bc,Bg}, Bd, {Be, Bi}}, which corresponds to five time slots to transmitsignals of the respective five beacon node groups. The grouping of thebeacon nodes described herein is merely examples, and other groupingsmay also be contemplated.

As illustrated by block 312, the method 300 may further comprisedetermining by the computing device, a schedule for the placed beaconnodes to be active. The schedule may be based at least in part on awaiting time of individual communication nodes to receive locationdetermination signals from one or more neighboring beacon nodes. Forexample, the schedule may assign time slots to individual beacon nodegroups determined earlier at block 308, for beacon nodes of a beaconnode group to transmit signals in the assigned time slots. In variousembodiments, the objective of the scheduling may be to permitcommunication nodes to estimate their respective distance measurementsfrom neighboring beacon nodes as frequent as possible. The objective mayalso be to reduce a waiting time for individual communication nodes toreceive signals from at least three unique neighboring beacon nodes.This may, for example, ensure that a communication node may receivesignals from at least three neighboring beacon nodes at rapid successionand determine its location, and may help determine locations of a mobilecommunication node with relatively more accuracy. Also, a communicationnode may have to be active for a less amount of time to receive signalsfrom the neighboring beacon nodes. In various embodiments, for theexample beacon node grouping previously discussed, an example broadcastordering may be {Ba, Bb, {Bc, Bg}, {Bh, Bi}, Bd, Be, Bf}, which mayresult in an average waiting time of three time slots for mostcommunication nodes located any where in the terrain 200. The schedulingof the beacon nodes will be discussed in more details further below.

Location Discovery (LD) Problem

As previously discussed with reference to FIG. 1, the locations of oneor more beacon nodes in a terrain may be known (e.g., once the beaconnodes are placed in the terrain in accordance with, for example, method300 of FIG. 3), and the location of one or more communication nodes inthe network may be estimated. In various embodiments, the locationdiscovery problem may be formulated using non-linear programming and/orlinear programming approach.

Non-linear Programming (NPL) Based Formulation

In various embodiments, the location discovery problem may be formulatedas an instance of non-linear programming, where the unknown variablesmay be the locations of one or more communication nodes. For example,S={S_(i) (X_(Si), Y_(Si))}, i=1, . . . , S_(N), may be a set of S_(N)communication nodes with unknown locations, and B={B_(j) (X_(Bj),Y_(Bj))}, j=1, . . . , N_(B) may be a set of N_(B) beacon nodes withknown locations, where N_(B)≧3.

As previously discussed, it may be possible to estimate the distanced_(ij) between a communication node Si and its neighboring beacon nodeBj. Error (ε_(ij)) in the measured distance between communication nodeSi and the beacon node Bj may be expressed as a difference between thedistance d_(ij) and the estimated Euclidean distance, which may be givenas:ε_(ij)=√{square root over ((X _(Bj) −X _(si))²+(Y _(Bj) −Y_(si))²)}{square root over ((X _(Bj) −X _(si))²+(Y _(Bj) −Y _(si))²)}−d_(ij)  Equation 1.

In various embodiments, an objective function may be to minimize orreduce a likelihood of location discovery errors or an appropriatefunction of the location discovery errors. In various embodiments, thelikelihood of location discovery error may refer to an estimated,probable or likely value of the location discovery error. The term“minimize” and/or the like as used herein may include a global minimum,a local minimum, an approximate global minimum, and/or an approximatelocal minimum. Likewise, it should also be understood that, the term“maximize” and/or the like as used herein may include a global maximum,a local maximum, an approximate global maximum, and/or an approximatelocal maximum. One or more techniques for formulation and solution of anonlinear objective function for estimating locations of one or morecommunication nodes may be well known to those skilled in the art, andmay be outside the scope of this disclosure.

Linear Programming (LP) Based Formulation

Linearization of a system of non-linear equations by squaring andsubtracting the equations from one another, and solving the linearequations by taking the minimum mean square estimate (MMSE) is a usefulmethod of linearizing problem sets. Note that the system of linearequations may be solved optimally by singular value decomposition (SVD)under the assumption that the ranging error model follows the Gaussiandistribution.

However, as discussed in more details herein later, in variousembodiments, the ranging errors of real deployed communication nodes maynot follow the Gaussian distribution. Instead, the ranging errors mayhave complex forms that cannot be captured by existing parametricdistributions. Thus, the problem of location discovery may be formulatedusing nonlinear programming, and the formulation may be linearized andsolved using linear programming (LP) and/or convex programmingtechniques.

During the linearization process, equation 1 may be simplified toobtain:d _(ij) ²+2d _(ij)ε_(ij)+ε_(ij) ² =X _(Bj) ²−2X _(Bj) X _(Si) +X _(Si) ²+Y _(Bj) ²−2Y _(Bj) Y _(Si) +Y _(Si) ²  Equation 2.

A number of such equations may be obtained (e.g., between thecommunication node Si and individual beacon nodes B1, B2, . . . , BK),and the equations may be manipulated to obtain a linear programmingbased formulation of the location discovery problem, as is well known tothose skilled in the art and is outside the scope of this disclosure.

Distance Measurement Error Model

In practice, due to a number of reasons (e.g., obstacles located betweentwo nodes, noise present in measurements, a large network size, a largenumber of nodes deployed in the network, interference, technology usedfor distance measurement, etc.), distance determination between twonodes may include inaccuracies. In various embodiments, a distancemeasurement error model may be constructed to model the errorprobability related with distance determination between any two nodes.

FIG. 4 illustrates an example method 400 for constructing a distanceerror measurement model, in accordance with various embodiments of thepresent disclosure. Similarly, various ones of these embodiments may beinstantiated on a computing device, that is, practicing the method on acomputing device provided with the decision logic to be described. Themethod 400 may be implemented in an experimental setup, where thelocations of one or more communication nodes may be known in advance toa user or a system setting up the experiment. The example method 400 mayinclude one or more functions or processing operations as illustrated byblocks 404, 408, 412, 416, 420, and/or 424.

At block 404, the location of one or more communication nodes in anetwork may be known (e.g., retrieved from a user or a systemcoordinating and/or controlling the experimental set up), in addition tothe known location of one or more beacon nodes in the network. Forexample, the location of one or more communication nodes in a networkmay be known in advance for at least the purpose of constructing such amodel in an experimental set up. In various embodiments, the locationinformation of the nodes may be received by the respective nodes.Alternatively, in various embodiments, a centralized computing device(not illustrated in FIG. 1), a user, and/or an administrator of thenetwork may receive the location information.

At block 408, from the known locations of any two neighboring nodes(e.g., a communication node and a beacon node), the actual distancebetween the two nodes may be calculated. At block 412, the distancebetween the two nodes may be estimated using one of many techniques,based at least in part on one of the nodes receiving signals from theother node. The actual and the estimated distances between the two nodesmay be compared to determine, at block 416, a distance measurement errorfor the two nodes. In various embodiments, this distance measurementerror may be an indication of accuracy of distance determination (ordistance estimation, since some amount of inaccuracy may be involved)between the two nodes.

At block 420, the method 400 may determine if all of the available pairsof neighborhood nodes have been considered. The process 400 may continuefrom block 420 to block 408, where the process may repeat for otherpossible pairs of neighboring nodes. Otherwise, the process 400 maycontinue from block 420 to block 424, where a distance error model maybe constructed based at least in part on the determined distancemeasurement error for possible pairs of neighboring nodes.

FIG. 5 illustrates an example distance measurement model 500, inaccordance with various embodiments of the present disclosure. The model500 includes a graph plotting a plurality of estimated distances (e.g.meters) and the associated measurement errors (e.g. meters) for anexample wireless network. The data in the figure may be from a wirelessnetwork similar to that in FIG. 1, but may employ a larger number ofcommunication nodes and beacon nodes (e.g., the total number ofcommunication nodes may be around 2000). These two sets of data may bethe input to the distance measurement error model.

As observed from model 500, for the example network, for a largermeasurement distance between any two nodes, the probability of error inthe measurement may be greater. That is, as distance between two nodesgrows larger, the distance measurement may be more prone to error, asindicated by larger number of scattered data points beyond, for example,at around 40 meter (m) along the x-axis. Also, nodes that may be furtherapart (e.g., more than at around 50 m apart) may be out of each other'ssignal range, and may not be able to exchange location and distancemeasurement information, resulting in substantially less number ofmeasurements beyond the 50 m range.

In various embodiments, the solid curves (e.g., curves 510, 512, etc.)in FIG. 5 may correspond to a percentage of the points that may form thelowest C % of the cumulative density function (CDF). Five such examplevalues of C are illustrated in FIG. 5. For example, the curve for C=75%may approximately have a form of 0.0004M³−0.0233M²+0.4478M−2.4591, whereM may be the distance measurement in meters. The individual curves maydescribe the probability of any given measurement that may have certainerror with C % confidence. For example, for a distance measurement ofaround 10 m, according to the curve C=75%, there may be approximately75% probability that the measurement may have an error of around 0.088 m(e.g. 0.0004(10)3−0.0233(10)2+0.4478(10)−2.4591). In another example,for a distance measurement of around 45 m, there may be a 75%probability that the measurement may have an error of around 6.959 m.

FIG. 6 illustrates another example distance measurement model 600, inaccordance with various embodiments of the present disclosure. The model600 of FIG. 6 illustrates an alternative view of the measurement model500 of FIG. 5. The x-axis of FIG. 6 may represent measurement errorvalues, and y-axis may illustrate the probability distribution function(PDF) for various example distance measurement. For example, FIG. 6includes plots 610 and 612 for distance measurement of 10 m and 45 m,respectively. As observed from plot 612, for a distance measurement of45 m, there may be a 75% probability that the measurement may have anerror of approximately 7 m.

Outdoor and Indoor Terrain Models

In various embodiments, estimation of distance between two nodes andsubsequent estimation of location of a communication node may be basedat least in part on the terrain in which the nodes may be deployed. Forexample, some or all the nodes may be deployed in an indoor terrainand/or an outdoor terrain, with one or more obstacles (e.g., trees,buildings, walls, etc.) in the environment that may hinder communicationbetween two nodes, as discussed with respect to FIG. 2. To moreaccurately depict the terrain in which the nodes may be deployed, anoutdoor and an indoor terrain model may be simulated that may bescalable both in terms of size and resolution and may mimic an actualterrain to the extent possible, and both the models may be parameterizedin terms of size, resolution, density and level of clustering of theobstacles.

FIG. 7 illustrates an example outdoor terrain model 700, in accordancewith various embodiments of the present disclosure. The outdoor terrainmodel 700 illustrates an example simulated outdoor terrain of size 1000m×1000 m. The simulated model may be divided in a number of grids (e.g.,grid 704) and the grey portions in the model 700 may depict obstacles(e.g., obstacles 710, 712 and 714), e.g., buildings, trees, buses, etc.In various embodiments, the outdoor terrain model may employ a number ofmodeling paradigms, including but not limited to statistical fractalsand interacting particles. Fractals may ensure that for individuallevels of resolution, the terrain may be statistically isomorphic. Also,interacting particles may be used as the mechanism to maintain aspecified density of obstacles and to create a specified level ofclustering in order to enforce self-similarity at different levels ofgranularity.

In various embodiments, the outdoor terrain model 700 may be createdusing an iterative procedure. The obstacles may be generated based atleast in part on an assumption that obstacles are mutually attracted. Invarious embodiments, the model may be generated iteratively on a finegrid, one obstacle at a time, till a user specified density may beachieved. During individual steps of the iterative process, the positionof the next grid cell may be selected proportional to the weight of thegrid according to a user specified cluster factor (CF). The outdoorterrain model 700 may be generated with CF=5. The weight of individualunoccupied grid cell may be the weighted sum of the impact of its eightneighboring grid cells. Occupied horizontal and vertical neighboringgrid cells may have an impact of √{square root over (CF)}, and diagonalneighboring grid cells may have an impact of CF, although any otherappropriate impact factor may be used for one or more neighboring cells.If the weight of an unoccupied cell exceeds a threshold level, theunoccupied cell may change to an occupied status. A cluster factor CF=1may result in uniformly distributed obstacles. For a relatively higherCF, the probability of forming one or more large obstacle may be higher.

FIG. 8 illustrates an example indoor terrain model 800, in accordancewith various embodiments of the present disclosure. The simulated indoorterrain model may be 1000 m×1000 m, and may include obstacles in theform of walls (e.g., wall 810). In various embodiments, the walls may begenerated by recursively partitioning the space in rectangles byfollowing a triangle shaped probability distribution function. A userspecified threshold may be used to prevent the walls from being placedtoo close to each other. For example, during the iterative generation ofthe model 800, any proceeding two partitions may have a user specifiedminimum threshold area ratio R. For example, for R=1/9, any partitionthat divides the space into ≦10% and ≧90% of its original size may notbe allowed. Although not illustrated in FIG. 8, the indoor model 800 mayalso include obstacles (other than walls), e.g., furniture, staircase,etc. and may be generated using the iterative process discussed withrespect to the outdoor terrain model.

Location Discovery (LD) Error Model

As previously discussed, the estimation of distance between two nodesmay have a probability of error, which may also introduce error in thelocation discovery (e.g., a difference between the actual location andan estimated location) of a communication node.

In various embodiments, it may be intended to generate a locationdiscovery (LD) error model that may correlate one or more attributes ofindividual communication nodes with the error in estimating the locationof respective communication nodes. For example, an LD model maycorrelate the number of neighboring nodes of individual communicationnodes with the error in estimating the location of the respectivecommunication nodes.

FIG. 9 a illustrates an example model 900 that includes correlationbetween the number of neighboring nodes (e.g., including neighboringcommunication nodes and/or neighboring beacon nodes) of individualcommunication nodes and the corresponding CDF values for a plurality(e.g., 1,000) communication nodes, in accordance with variousembodiments of the present disclosure. In various embodiments, the model900 may be generated using a simulated scenario, where 1,000communication nodes may be randomly deployed in an example terrain.

FIG. 9 b illustrates an example location discovery error model 920 thatincludes correlation between the CDF values of number of neighboringnodes of individual communication nodes and the corresponding locationestimation error for 1,000 communication nodes, in accordance withvarious embodiments of the present disclosure. The model 920 alsoillustrates least squares fit trend lines, along with corresponding R²values.

As will be readily understood in light of the present disclosure, invarious embodiments, the models of FIGS. 9 a and 9 b may be generatedthrough simulation, at least partially in line with the methodpreviously discussed with respect to generation of the distancemeasurement error model.

In various embodiments, it may be possible to estimate, from models 900and 920, an average location estimation error of a communication node ifthe number of neighboring nodes of such a communication node is known.It may be observed from FIGS. 9 a and 9 b that as the average number ofneighboring nodes may increase, the corresponding resultant locationestimation error may decrease. As the CDF values of FIGS. 9 a and 9 bcorrelate the number of neighboring nodes of individual communicationnode with average location estimation error of the respectivecommunication nodes, the CDF values will henceforth be denoted byCDF_(N).

Although not illustrated in FIGS. 9 a and 9 b, other attributes ofindividual communication nodes may also be correlated with thecorresponding location estimation error to create other locationdiscovery error models. For example, it may be possible to generate alocation discovery error model that may correlate the number ofneighboring beacon nodes of individual communication nodes (or acorresponding CDF value) with average location estimation error of therespective communication nodes, and the associated CDF may be referredas CDF_(BN) henceforth. For example, a communication node with highernumber of neighboring beacon nodes may have a lower average locationdiscovery error.

In various embodiments, the average location discovery error of acommunication node may also be based at least in part on the thirdlargest angle of its neighbors. FIG. 10 a illustrates an examplewireless network 1000 in accordance with various embodiments of thepresent disclosure. In the wireless network 1000, the angles formed bythe neighboring nodes of the communication nodes Sc may be given by A°,B°, C°, and D°. As illustrated, angle A° may be the third largest angleof the four angles. Thus, for a node Sc, the angles between all possiblepairs of neighbors of Sc with respect of Sc may be calculated, and thirdlargest angle A° may be determined accordingly. For example, assume aset of nodes S={Si}, i=1, . . . , N_(S). For individual node SiεS, thethird largest angle Ai° may be determined.

FIG. 10 b illustrates an example model 1020 that includes correlationbetween the third largest angle of neighboring nodes of individualcommunication nodes and the corresponding CDF values for a plurality(e.g., 1,000) of communication nodes, in accordance with variousembodiments of the present disclosure. FIG. 10 c illustrates an examplelocation discovery error model 1040 that includes correlation betweenthe CDF values of the third largest angle of neighboring nodes ofindividual communication nodes and the corresponding location estimationerror for a plurality (e.g., 1,000) of communication nodes, inaccordance with various embodiments of the present disclosure. Themodels 1020 and 1040 may measure how collinear a communication nodesnode's neighbors may be, and the communication nodes that may berelatively well surrounded may have a higher probability of beinglocated relatively more accurately than others. For example, if acommunication node has three neighbors that may be relatively far apartand spread out (e.g., relatively triangulated neighbors), this maycreate a relatively more restrictive solution space for the unknowncommunication node's location when compared to a communication node thathas nearly collinear neighboring nodes.

In various embodiments, the models of FIGS. 9 a and 9 b may be generatedthrough simulation by determining, for individual nodes Si in the set ofcommunication nodes S={Si}, i=1, . . . , N_(S), the corresponding thirdlargest angle Ai°. Subsequently, the CDF plot of FIG. 10 b may begenerated based on Ai, i=1, . . . , N_(S), wherein for any givencommunication node with third largest angle Ai, the corresponding CDFvalue may be denoted by CDF_(Angle)(Si) or as CDF_(A)(Si). Subsequently,individual communication node Si's relative location errors ε_(i), i=1,. . . , N_(S), may be correlated against CDF_(A)(Si), as illustrated inthe plot in FIG. 10 c. FIG. 10 c also illustrates least square fit trendlines, along with corresponding R2 values.

FIG. 11 illustrates an example location discovery error model 1100 thatincludes correlation between the third largest angle of neighboringnodes of individual communication nodes, the number of neighboring nodesof corresponding communication nodes, and the location estimation errorof the corresponding communication nodes, in accordance with variousembodiments of the present disclosure. In various embodiments, the 3-Dplot in model 1100 may be generated by combining the 2-D plots of FIGS.9 a, 9 b, 10 a and 10 b. From model 1100, an average location estimationerror of a communication node may be determined if the third largestangle of neighboring nodes and the number of neighboring nodes of such acommunication node is known.

Although not illustrated in the figures, other attributes of individualcommunication nodes may be correlated with the corresponding locationestimation error. For example, for a lower average distance measurementto three closest beacon nodes from a communication node, the averagelocation discovery error may be lower for such a communication node. Invarious embodiments, for a communication node that may have a highertotal hop count to other nodes in the network from the communicationnode, the average location discovery error for such a communication nodemay be higher. A LD model, along with a plot illustrating correlation ofthese attributes of individual communication nodes with thecorresponding location estimation error, may be generated in line withFIGS. 9 a, 9 b, 10 a and/or 10 b.

LDI:Beacon Node Placement

As discussed earlier with respect to FIG. 3, in various embodiments, oneor more beacon nodes may form an LDI, which may aid in determininglocations of one or more communication nodes in a network. Thus, an LDImay include a relatively small number of beacon nodes in a terrain thatmay permit any arbitrary communication node in the terrain to locateitself. In various embodiments, beacon placement may be a part ofbuilding an LDI, and the beacon placement problem may includedetermining the minimum number and position of beacon nodes required fora known terrain, such that one or more communication nodes placed at anylocation in the terrain may receive signals from at least threeneighboring beacon nodes, to permit location estimation of the one ormore communication nodes.

FIG. 12 illustrates an example method 1200 for determining a pluralityof locations in a terrain to place a corresponding plurality of beaconnodes, in accordance with various embodiments of the present disclosure.Various ones of these embodiments may be instantiated on a computingdevice, that is, practicing the method on a computing device providedwith the decision logic to be described. In various embodiments, themethod 1200 may include one or more of functions or processingoperations illustrated by blocks 1204, 1208 and/or 1212.

At block 1204, one or more of a plurality of communication nodes may begrouped into a plurality of communication node groups, such that the allcommunication nodes in individual communication node groups may be inproximity with each other and may not have any obstacles in between. Invarious embodiments, the grouping in block 1204 may be performed byrandomly generating a plurality of communication nodes in a knownterrain. The communication nodes may be generated, for example, suchthat the generated communication nodes may not overlap with a knownobstacle. The location of the generated communication node may be knownto the respective communication nodes or to a centralized object (e.g.,a server, a user of the network, a computing device, or the like) of thenetwork.

The grouping at block 1204 may be performed, for example, such that allcommunication nodes in individual communication node groups may be inclose proximity, without any obstacles in between, and all suchcommunication nodes may have one or more common beacon nodes asneighboring nodes. In various embodiments, the grouping may be performedusing information about the terrain.

In various embodiments, block 1204 may also include selecting arespective representative communication node from respective individualcommunication node groups to form a dominating set of representativecommunication nodes. For at least the purpose of building an LDI andunless otherwise mentioned, the representative communication node of acommunication node group may represent other communication nodes in thecommunication node group. In various embodiments, the number ofrepresentative communication nodes in the dominating node may besubstantially equal to the number of communication node groups. Invarious embodiments, the representative communication node of acommunication node group may have one or more properties that may besimilar to other communication nodes in the communication node group(e.g., having one or more common neighboring beacon nodes, being inclose proximity with communication nodes in the group, etc.).

In various embodiments, the grouping of the communication nodes andformation of the domination set may be performed using, for example,integer linear programming (ILP). For example, S={S_(i)}, i=1, . . . ,N_(S), may be the plurality of communication nodes that may be dividedin N_(DS) number of groups so that the dominating set DS={DS_(i)}, i=1,. . . , N_(DS). In various embodiments, a constant E_(ij) may denotewhether nodes Si and Sj share the same set of three beacon nodes, and avariable may di denote whether node Si is selected in the dominating set(i.e., whether Si is a representative communication node). An objectivefunction may be formulated to generate the communication node groups andselect the dominating set as follows:

$E_{ij} = \left\{ {{\begin{matrix}{1,} & \begin{matrix}{{{{if}\mspace{14mu}{communication}\mspace{14mu}{nodes}\mspace{14mu}{Si}}\&}{Sj}\mspace{14mu}{receive}\mspace{14mu}{acoustic}} \\{{{signals}\mspace{14mu}{from}\mspace{14mu}{the}\mspace{14mu}{same}\mspace{14mu}{set}\mspace{14mu}{of}\mspace{14mu}{three}\mspace{14mu}{beacons}}\;}\end{matrix} \\{0,} & {otherwise}\end{matrix}d_{i}} = \left\{ {{\begin{matrix}{1,} & \begin{matrix}{{if}\mspace{14mu}{communication}\mspace{14mu}{node}\mspace{14mu}{Si}\mspace{14mu}{is}\mspace{14mu}{selected}\mspace{14mu}{in}\mspace{14mu}{the}} \\{{dominating}\mspace{14mu}{set}}\end{matrix} \\{0,} & {otherwise}\end{matrix}{OF}\text{:}\min{\sum\limits_{i = 1}^{N_{S}}{d_{i}{such}\mspace{14mu}{{that}:{{\sum\limits_{i = 1}^{N_{S}}{d_{j}E_{ij}}} \geq 1}}}}},\mspace{14mu}{i = 1},\ldots\mspace{14mu},{N_{S}.}} \right.} \right.$

The objective function, thus, may minimize and/or reduce the number ofrepresentative communication nodes in the dominating set (e.g., minimizeand/or reduce the number of communication node groups). The constraintmay ensure that for any individual communication node Si, either Si maybe in the dominating set or there may be at least one communication nodeSj in the dominating set, such that Si and Sj may receive signals fromthe same set of three neighboring beacon nodes (e.g., E_(ij)=1, and Sjmay be the representative communication node, representing communicationnode Si in the dominating set).

The method 1200 may further include, at block 1208, formulating anobjective function based at least in part on one or more attributesbetween individual representative communication nodes and one or morebeacon nodes that are to be placed in the terrain. In variousembodiments, the objective function may be formulated using non-linearprogramming. The unknown variables may be the coordinates of the set ofbeacon nodes B={Bi}, i=1, . . . , NB, (with coordinates (X_(Bi),Y_(Bi)), i=1, . . . , NB) that are to be placed in the terrain. Invarious embodiments, the objective function may reduce and/or minimizean expected location discovery error of representative communicationnodes based at least in part on one or more attributes between theindividual representative communication nodes and one or more beaconnodes in the set B.

An example objective function formulated in block 1208 may be

$\begin{matrix}{{{{OF}\text{:}\min\; F_{1}} = {\sum\limits_{i = 1}^{N_{DS}}{M_{i}\left( {{CDF}_{N}\left( {DS}_{i} \right)} \right)}}},} & {{Equation}\mspace{14mu} 3}\end{matrix}$where the function F1 may be based at least in part on a locationdiscovery error model that may correlate the number of neighboring nodes(or the associated CDF_(N) value) of individual representativecommunication nodes and respective location discovery error (e.g., frommodel 920 of FIG. 9 b). For example, CDF_(N)(DSi) may be the CDF valueassociated with the number of neighboring nodes of the representativecommunication node DSi in the dominating set DS, where the CDF_(N)(DSi)may be determined from, for example, model 900 of FIG. 9 a. Function M₁may map the CDF_(N)(DSi) to the corresponding average location discoveryerror for the communication node DSi and may be determined, for example,from the location discovery error model 920 of FIG. 9 b.

In various embodiments, individual representative communication nodesmay have at least three neighboring beacon nodes. Accordingly, anadditional term F2 may be added to equation 3, where F2 may be equal toM₂(CDF_(BN)(DSi))), where CDF_(BN) may be the CDF value associated withthe number of neighboring beacon nodes of the representativecommunication node DSi in the dominating set DS. Determination ofCDF_(BN) may be performed in a manner similar to determination ofCDF_(N), as has been discussed herein earlier. Function M₂ may map theCDF_(BN)(DSi) to the corresponding average location discovery error forthe communication node DSi. Accordingly, a modified objective functionmay be:OF:min F=W·F ₂ +F ₁  Equation 4,where W may be an appropriate weighing factor that may emphasize F2 toensure that individual representative communication nodes may have atleast three neighboring beacon nodes, prior to considering minimizing orreducing the expected location error (e.g. F1).

In various embodiments, the objective function of equation 4 may beextended to include additional attributes between individualrepresentative communication nodes and one or more beacon nodes that areto be placed in the terrain. For example, an additional term may beincluded in the objective function to take into account the thirdlargest angle of all neighboring nodes of a representative communicationnode (as determined, for example, in FIGS. 10 a-10 c and the previouslydescribed CDF_(A) value).

The method 1200 may further include, at block 1212, solving theobjective function to determine a minimum required number of beaconnodes for the terrain, and locations in the terrain to place the minimumnumber of beacon nodes. The solving may be performed using one or moreavailable non-linear programming tools or software. For example, thesolving may be performed using publicly available optimization softwareWNLIB, which is an ANSI C subroutine library that includes numericalroutines such as conjugate-gradient and conjugate-directions, simplexmethod, etc. and available at http://www.willnaylor.com/wnlib.html.Other non-linear programming tools and programs may also be used for thesolving at block 1212.

LDI:Beacon Node Assignment

As discussed earlier with respect to block 308 of method 300 of FIG. 3,after the placement of the beacon nodes in the terrain may bedetermined, the beacon nodes may be divided in one or more beacon nodegroups, based at least in part on the placed beacon nodes withinindividual beacon node groups being able to be active simultaneouslywithout causing interference or collision at the individualcommunication nodes of the plurality of communication nodes. An examplecriterion for grouping the beacon nodes may be that a pair of beaconnodes Bi and Bj may not transmit simultaneously if the pair causes anyreceiving communication node to have collision. For example, if at leastone communication node Sa receives transmission from the both beaconnodes Bi and Bj, then the beacon nodes Bi and Bj may not transmitsimultaneously (e.g., may be placed in separate beacon node groups) toavoid chances of collision or interference at the receivingcommunication node Sa.

FIG. 13 illustrates an example method 1300 for determining a pluralityof beacon node groups for the placed beacon nodes, in accordance withvarious embodiments of the present disclosure. Various ones of theseembodiments may be instantiated on a computing device, that is,practicing the method on a computing device provided with the decisionlogic to be described. In various embodiments, the beacon nodes may begrouped based at least in part on the placed beacon nodes withinindividual beacon node groups being able to be active simultaneouslywithout causing interference or collision at the individualcommunication nodes of the plurality of communication nodes. In variousembodiments, the method 1200 may include one or more of functions orprocessing operations illustrated by blocks 1304, 1308 and/or 1312.

At block 1304, beacon node pairs that may not transmit simultaneouslyare identified. For example, if a communication node Sa receives signalsfrom beacon nodes Bi and Bj and uses the received signals from beaconnodes Bi and Bj to determine the location of the communication node Sa,then the beacon node pair Bi and Bj may not transmit simultaneously. Invarious embodiments, for individual pairs of beacon nodes, it may bedetermined whether the pair of beacon nodes may transmit simultaneously.In various embodiments, binary variable Rij may represent a userspecified condition that may restrain the pair of beacon nodes Bi and Bjfrom transmitting simultaneously. For example,

$R_{ij} = \left\{ \begin{matrix}{1,} & {{if}\mspace{14mu}{beacons}\mspace{14mu}{Bi}\mspace{14mu}{and}\mspace{14mu}{Bj}\mspace{14mu}{may}\mspace{14mu}{not}\mspace{14mu}{transmit}\mspace{14mu}{simultaneously}} \\{0,} & {otherwise}\end{matrix} \right.$In various embodiments, R_(ij) may be determined for i=1, . . . , N_(B),and j=1, . . . , N_(B), where N_(B) may be the total number of beaconnodes to be placed in the terrain.

The method 1300 may include, at block 1308, formulating the problem ofgrouping the placed beacon nodes as a graph coloring problem. Graphcoloring is a graph labeling formulation used in graph theory, integerlinear programming, or the like. In various embodiments, during theformulation at block 1304, individual beacon nodes may, for example,represent a node or vertex of a graph. An edge between any two verticesin the graph may represent a condition associated with the associatedbeacon nodes transmitting signals simultaneously. For example, if Rij=1(e.g., if beacon nodes Bi and Bj may not transmit simultaneously), thenthe edge connecting the i^(th) and j^(th) vertices (representing beaconnodes Bi and Bj, respectively) of the graph may be approximately equalto 1.

The method 1300 may include, at block 1312, solving the graph coloringproblem to determining a grouping of the beacon nodes. In variousembodiments, the graph coloring problem may be solved using toolsavailable in graph theory. In various embodiments, the graph coloringproblem may be solved to determine, for example, a minimum number ofcolors (e.g., T number of colors) needed to color the entire graph,wherein individual colors may represent respective beacon node groupsBG={BGi}, i=1, . . . , T. Thus, the minimum number of colors utilizedmay represent the minimum number of beacon node groups. The minimumnumber of colors utilized may also represent the minimum total number oftime slots that may be used for all the beacon nodes to transmitacoustic signals at least once (e.g., one time slot for a respectivebeacon node group). For individual colors, the associated vertices(e.g., associated beacon nodes) may also be identified, which mayrepresent the beacon nodes included in the respective beacon nodegroups. That is, the beacon nodes that may be colored by the same colormay form a beacon node group, and these beacon nodes may transmitsignals simultaneously.

Although the beacon node grouping in method 1300 may be solved usinggraph coloring, any other appropriate integer linear programming toolsmay also be used (instead of and/or in addition to graph coloring) forgrouping the beacon nodes, as will be readily understood based on theteachings provided in this disclosure.

LDI:Beacon Node Broadcast Scheduling

As discussed earlier with respect to block 312 of method 300 of FIG. 3,after the placement of the beacon nodes in the terrain and the groupingof the beacon nodes may be determined, a schedule for activating thebeacon node groups may also be determined, based at least in part on awaiting time of a communication node to receive location determinationsignals from a plurality of beacon nodes. For example, as earlierdiscussed, there may be T number of beacon node groups BG={BGi}, i=1, .. . , T, with individual beacon node groups being assigned a respectivetime slot t (where 1≦t≧T). For individual round of transmissions, beaconnodes in a beacon node group may transmit signals in the time slotassigned to the group. Thus, each round of transmission may include Ttime slots (one for respective beacon node group). For example, thet^(th) beacon node group BG_(t) may transmit at time slots t, (t+T),(t+2T), . . . and so on.

In various embodiments, N_(DS) may represent the number ofrepresentative communication nodes in the dominating set DS, so thatindividual representative communication nodes in the dominating set DSis represented by {DSi}, i=1, . . . , N_(DS). For individualrepresentative communication nodes DSi, constant Fi (i=1, . . . ,N_(DS)) may represent the group of communication nodes that DSi mayrepresent. That is, DSi be the representative communication node of thecommunication node group Fi (e.g., see block 1204 of method 1200), andFi may represent the group of communication nodes that, along withrepresentative communication node DSi, may have a common set ofneighboring beacon nodes. In various embodiments, BFi={BGi1, BGi2,BGi3}, where BGi1, BGi2, BGi3 may be three beacon node groups thatinclude three respective beacon nodes that may serve the group ofcommunication nodes Fi, i=1, . . . , N_(DS). Thus, BFi may represent theset of beacon node groups that may include beacon nodes that serveindividual communication nodes of the communication node group Fi. Forexample, BF2={5, 2, 8} may imply that the group of communication nodesF2 may rely on three neighboring beacon nodes that belong to beacon nodegroups 5, 2 and 8, respectively. In various embodiments, integervariable t[BGij] may denote the time slot in which the beacon node groupBGij may transmit, 1≦t[BGij]≦T.

In various embodiments, one of the criteria used for determining abeacon broadcast schedule may be to reduce or minimize the powerconsumption of one or more communication nodes that may receive signalstransmitted by the beacon nodes. For example, one of the objectives maybe to reduce or minimize a time individual communication node's receivermay have to be active, i.e. the time of receiving the acoustic signalsfrom three beacon nodes.

For example, in an example situation, for individual rounds of beacontransmission, there may be 9 time slots (e.g., for 9 different beaconnode groups). During a first example scenario, a group of communicationnode (e.g., communication node group F2) may receive transmission fromneighboring beacon nodes at time slots 1, 2, and 5 (and may keep onreceiving signals in time slots 10, 11, 14, 19, 20, 23, etc. duringsubsequent rounds of signal transmission). Individual communicationnodes in the communication node group F2 may be active (e.g., enabled toreceive signals from neighboring beacon nodes) during time slots 1-5(and also during time slots 10-14, 19-23, etc. subsequent rounds ofsignal transmission), and may be inactive (e.g., configured not toreceive any beacon node signals) during time slots 6-9 (and also duringtime slots 15-18, 24-28, etc. subsequent rounds of signal transmission).For the purpose of this disclosure and unless otherwise specified, amaximum active period of a communication node may refer to a maximumnumber of consecutive time slots the communication node may remainactive for receiving signals from neighboring beacon nodes. For example,if a communication node uses signals received from M (e.g., 3) beaconnodes (B1, B2, . . . , BM) for determining the location of thecommunication node and the communication node receives the signals inthat order (B1, B2, . . . , BM), the maximum active period of thecommunication node may include the time slot when a first beacon nodesignal may be received (e.g., B1), the time slot when a last beacon nodesignal may be received (e.g., BM), and any intervening time slots inbetween the two time slots (e.g., when signals from B2, . . . , B(M−1)are received). For example, for the first example scenario, the maximumactive period of communication nodes in group F2 may be 5 time slots.

During a second example scenario, the communication node group F2 mayreceive transmission from neighboring beacon nodes at time slots 2, 3,and 4 for each round of beacon node transmission (and also during timeslots 11, 12, 13, 20, 21, 22, etc.). Hence, individual communicationnodes in the communication node group F2 may be active during time slots2-4 (and also during 11-13, 20-22, etc.), and may be inactive duringtime slots 4-10 (and also during 14-19, 23-28, etc.). In the secondexample scenario, the individual communication nodes in thecommunication node group F2 may have to remain active for less time ascompared to the first example scenario. For the second example scenario,the maximum active period of communication nodes in group F2 may be 3time slots.

In various embodiments, a beacon broadcasting schedule may be developedthat may reduce, minimize and/or optimize the time the individualcommunication nodes may have to be active to receive acoustic signalsfrom neighboring beacon nodes.

In various embodiments, signals from three beacon nodes from respectivethree beacon node groups BGi1, BGi2, BGi3 may arrive at communicationnodes included in a communication node group Fi in the same order (e.g.,BGi1, BGi2, BGi3). An objective function may be given as:OF:min H ₁=max{|t[BG _(i1) ]−t[BG _(i2) ]|,|t[BG _(i1) ]−t[BG _(i3)]|,|t[BG _(i2) ]−t[BG _(i3) ]|}, i=1, . . . , N _(DS)  Equation 5.In various embodiments, the maximum (e.g., “max”) function may beimplemented by introducing another variable v, whereinv≧|t[BG_(i1)]−t[BG_(i2)]|, v≧|t[BG_(i1)]−t[BG_(i3)]|, andv≧|t[BG_(i2)]−t[BG_(i3)]|. The variable v may be subject tominimization. In various embodiments, the objective function may be toreduce or minimize the maximum active period of the communication nodes.

In various embodiments, the communication nodes in the group ofcommunication nodes Fi may receive transmission from beacon nodesincluded in the beacon node groups BGi1, BGi2, BGi3 in any other orderas well (e.g., BGi2, BGi1, BGi3). Accordingly, there may be sixadditional sets of equations (that may be of the same form as equation5), in which either any one or two beacon node groups (in anycombination) may have the total number of time slots T added. Forexample,H ₂=max{|(t[BG _(i1) ]+T)−t[BG _(i2)]|,|(t[BG _(i1) ]+T)−t[BG _(i3)]|,|t[BG _(i2) ]−t[BG _(i3) ]|}, i=1, . . . , N _(DS)  Equation 6.

Equation 6 (i.e., the term H₂) may be associated with an examplesituation where signals are received from beacon nodes belonging tobeacon node groups BGi2, BGi3, BGi1, in that order. Another exampleequation in which the term T may be added to two beacon node groups maybe given as:H ₃=max{|(t[BG _(i1) ]+T)−t[BG _(i2)]|,|(t[BG _(i1) ]+T)−(t[BG _(i3)]+T)|,|t[BG _(i2)]−(t[BG _(i3) ]+T)|}, i=1, . . . , N _(DS)  Equation 7.

Let Hi, i=1, . . . , 7 denote the maximum of the individual equations.In various embodiments, an objective function may be given by:OF:min{H₁, H₂, . . . , H₇}  Equation 8.

Thus, equation 8 may minimize or reduce a waiting time (i.e., the time acommunication node may have to be active and waiting for beaconsignals), and/or minimize or reduce a maximum active period ofindividual communication nodes. In various embodiments, a modifiedversion of the objective function may reduce or minimize an averagewaiting time individual representative communication nodes in thedominating set DSi (and therefore, the communication nodes that therepresentative communication node DSi represents) have to wait in orderto receive acoustic signals from at least three neighboring beaconnodes. In various embodiments, it may also be possible to minimizeenergy from the beacon node's perspective, by reducing or minimizing anumber of times each beacon node transmits, while satisfying theconstraints.

In various embodiments, equations 5-8 may be solved using integer linearprogramming, graph coloring, or the like, known to those skilled in theart.

FIG. 14 illustrates an example method 1400 for scheduling signaltransmission of a plurality of beacon nodes, in accordance with variousembodiments of the present disclosure. Various ones of these embodimentsmay be instantiated on a computing device, that is, practicing themethod on a computing device provided with the decision logic to bedescribed. The method 1400 may include one or more function orprocessing operations as illustrated by blocks 1404, 1408 and/or 1412.

At block 1404, formulating an objective function (e.g., objectivefunctions of equations 5, 6, 7 and/or 8) based at least in part onwaiting times of individual communication nodes for receiving locationdetermination signals from a plurality of beacon nodes. In variousembodiments, the objective function may reduce or minimize the waitingtime. In various embodiments, the objective function reduce or minimizethe maximum active periods of individual communication nodes, whereinthe maximum active period of a communication node may be the time acommunication node remains active for receiving signals from neighboringbeacon nodes.

The method 1400 may include, at 1408, solving the objective function todetermine a schedule for signal transmission of the plurality of beaconnodes. In various embodiments, the objective function may be solved, forexample, using integer linear programming, graph coloring, or the like.

Computing System

FIG. 15 illustrates an example computing system 1500 that may besuitable for practicing various embodiments of the present disclosure.Computing system 1500 may comprise processor 1510 and memory 1520.Computing system 1500 may include one or more data models and/orcomputation modules configured to practice one or more aspects of thisdisclosure. For example, the computing system 1500 may include a LDIcreation module 1530 that may be used to create a location discoveryinfrastructure for an example wireless network in a known terrain (e.g.,terrain 200 of FIG. 2). In various embodiments, the LDI creation module1530 may include one or more additional modules, including a beacon nodeplacement module 1534 that may be arranged to determine a number andlocations of beacon nodes to be placed in a terrain, based at least inpart on likely location discovery error for discovering locations of oneor more of the plurality of communication nodes using the placed beaconnodes. In various embodiments, the LDI creation module 1530 may alsoinclude a beacon node grouping module 1536 that may be arranged todetermine a plurality of beacon node groups for the placed beacon nodes,based at least in part on the placed beacon nodes within individualbeacon node groups being able to be active simultaneously withoutcausing interference or collision at individual communication nodes ofthe plurality of communication nodes. In various embodiments, the LDIcreation module 1530 may include also a beacon node scheduling module1538 that may be arranged to determine a schedule for the placed beaconnodes to be active, based at least in part on waiting time of individualcommunication nodes to receive location determination signals from aplurality of beacon nodes.

In various embodiments, the computing system 1500 may also include amodel construction and validation module 1550 that may be arranged toconstruct and/or validate one or more models (e.g., LD error model,indoor and/or outdoor terrain model, distance measurement error model,etc.) previously disclosed in this disclosure. In various embodiments,the computing system 1500 may also include a location discovery module1552, which may be arranged to discover locations of one or morecommunication nodes in a network, as previously discussed in thisdisclosure.

In various embodiments, the computing system 1500 may be coupled to anetwork 1564. In various embodiments, processor 1510 may be ageneral-purpose processor and memory 1520 may be a hard drive,solid-state drive, Random Access Memory (RAM), or other appropriate typeof memory. In various embodiments, a plurality of programminginstructions may be stored within memory 1520 or other memory (e.g.,external storage facility 1560) and configured to program processor 1510to function as described within this disclosure. In various embodiments,processor 1510 may be an Application Specific Integrated Circuit (ASIC),a field programmable gate array (FPGA), or other logic device havingspecific functions built or programmed directly into it.

In various embodiments, one or more modules (but not all) may be presentin the computing system 1500. In various embodiments, the computingsystem 1500 may be included in a communication node of a network. Insuch embodiments, the computing system 1500 may utilize the locationdiscovery module 1552, which may be arranged to discover a location ofthe communication node. In various embodiments, the computing system1500 may be included in a centralized computing device (e.g., a server)of a network or may be controlled by a user and/or an administrator ofthe network, and may be arranged to utilize the model construction andvalidation module 1550 to construct and/or validate one or more models,utilize the LDI module 1530 to develop an LD infrastructure of anetwork, and/or utilize the location discovery module 1552 to discoverlocations of one or more communication nodes.

FIG. 16 illustrates an example computing program product 1601 inaccordance with various embodiments, all arranged in accordance with thepresent disclosure. In various embodiments, computing program product1601 may comprise a signal bearing medium 1603 having programminginstructions stored therein. The signal bearing medium 1603 may be, forexample, a compact disk (CD), a digital versatile disk (DVD), asolid-state drive, a hard drive, computer disks, flash memory, or otherappropriate type of data/instruction storage medium. The computerprogramming product 1601 may be, for example, included in or employed toprogram or configure a communication node, a beacon node, and/or in apeer or centralized device in a wireless network. Embodiments are notlimited to any type or types of computing program products.

Signal bearing medium 1603 may contain one or more instructions 1605configured to practice one or more aspects of the disclosure.Embodiments may have some or all of the instructions depicted in FIG.16. Embodiments of computing program product 1601 may have otherinstructions in accordance with embodiments described within thisspecification. In various embodiments, the one or more instructions 1605may include one or more instructions for creating a LDI for estimatinglocations of one or more of a plurality of communication nodes in aterrain. In various embodiments, the one or more instructions 1605 mayinclude one or more instructions for determining a plurality oflocations in the terrain to place a corresponding plurality of beaconnodes, based at least in part on likely location discovery error fordiscovering locations of one or more of the plurality of communicationnodes using the placed beacon nodes. In various embodiments, the one ormore instructions 1605 may include one or more instructions fordetermining a plurality of beacon node groups for the placed beaconnodes, based at least in part on the placed beacon nodes withinindividual beacon node groups being able to be active simultaneouslywithout causing interference or collision at individual communicationnodes of the plurality of communication nodes. In various embodiments,the one or more instructions 1605 may include one or more instructionsfor determining a schedule for the placed beacon nodes to be active,based at least in part on waiting time of individual communication nodesto receive location determination signals from a plurality of beaconnodes.

In various embodiments, the signal bearing medium 1603 may include acomputer readable medium 1607, including but not limited to a CD, a DVD,a solid-state drive, a hard drive, computer disks, flash memory, orother appropriate type of computer readable medium. In variousembodiments, the signal bearing medium 1603 may also include arecordable medium 1609, including but not limited to a floppy disk, ahard drive, a CD, a DVD, a digital tape, a computer memory, a flashmemory, or other appropriate type of computer recordable medium. Invarious embodiments, the signal bearing medium 1603 may include acommunications medium 1611, including but not limited to a fiber opticcable, a waveguide, a wired or wireless communications link, etc.

Claimed subject matter is not limited in scope to the particularimplementations described herein. For example, some implementations maybe in hardware, such as employed to operate on a device or combinationof devices, for example, whereas other implementations may be insoftware and/or firmware. Likewise, although claimed subject matter isnot limited in scope in this respect, some implementations may includeone or more articles, such as a storage medium or storage media. Thisstorage media, such as CD-ROMs, computer disks, flash memory, or thelike, for example, may have instructions stored thereon, that, whenexecuted by a system, such as a computer system, computing platform, orother system, for example, may result in execution of a processor inaccordance with claimed subject matter, such as one of theimplementations previously described, for example. As one possibility, acomputing platform may include one or more processing units orprocessors, one or more input/output devices, such as a display, akeyboard and/or a mouse, and one or more memories, such as static randomaccess memory, dynamic random access memory, flash memory, and/or a harddrive.

Reference in the specification to “an implementation,” “oneimplementation,” “some implementations,” or “other implementations” maymean that a particular feature, structure, or characteristic describedin connection with one or more implementations may be included in atleast some implementations, but not necessarily in all implementations.The various appearances of “an implementation,” “one implementation,” or“some implementations” in the preceding description are not necessarilyall referring to the same implementations. Moreover, when terms orphrases such as “coupled” or “responsive” or “in response to” or “incommunication with”, etc. are used herein or in the claims that follow,these terms should be interpreted broadly. For example, the phrase“coupled to” may refer to being communicatively, electrically and/oroperatively coupled as appropriate for the context in which the phraseis used.

In the preceding description, various aspects of claimed subject matterhave been described. For purposes of explanation, specific numbers,systems and/or configurations were set forth to provide a thoroughunderstanding of claimed subject matter. However, it should be apparentto one skilled in the art and having the benefit of this disclosure thatclaimed subject matter may be practiced without the specific details. Inother instances, well-known features were omitted and/or simplified soas not to obscure claimed subject matter. While certain features havebeen illustrated and/or described herein, many modifications,substitutions, changes and/or equivalents will now, or in the future,occur to those skilled in the art. It is, therefore, to be understoodthat the appended claims are intended to cover all such modificationsand/or changes as fall within the true spirit of claimed subject matter.

There is little distinction left between hardware and softwareimplementations of aspects of systems; the use of hardware or softwareis generally (but not always, in that in certain contexts the choicebetween hardware and software may become significant) a design choicerepresenting cost vs. efficiency tradeoffs. There are various vehiclesby which processes and/or systems and/or other technologies describedherein may be effected (e.g., hardware, software, and/or firmware), andthat the preferred vehicle will vary with the context in which theprocesses and/or systems and/or other technologies are deployed. Forexample, if an implementer determines that speed and accuracy areparamount, the implementer may opt for a mainly hardware and/or firmwarevehicle; if flexibility is paramount, the implementer may opt for amainly software implementation; or, yet again alternatively, theimplementer may opt for some combination of hardware, software, and/orfirmware.

The foregoing detailed description has set forth various embodiments ofthe devices and/or processes via the use of block diagrams, flowcharts,and/or examples. Insofar as such block diagrams, flowcharts, and/orexamples contain one or more functions and/or operations, it will beunderstood by those within the art that individual function and/oroperation within such block diagrams, flowcharts, or examples may beimplemented, individually and/or collectively, by a wide range ofhardware, software, firmware, or virtually any combination thereof. Inone embodiment, several portions of the subject matter described hereinmay be implemented via Application Specific Integrated Circuits (ASICs),Field Programmable Gate Arrays (FPGAs), digital signal processors(DSPs), or other integrated formats. However, those skilled in the artwill recognize that some aspects of the embodiments disclosed herein, inwhole or in part, may be equivalently implemented in integratedcircuits, as one or more computer programs running on one or morecomputers (e.g., as one or more programs running on one or more computersystems), as one or more programs running on one or more processors(e.g., as one or more programs running on one or more microprocessors),as firmware, or as virtually any combination thereof, and that designingthe circuitry and/or writing the code for the software and or firmwarewould be well within the skill of one of skill in the art in light ofthis disclosure. In addition, those skilled in the art will appreciatethat the mechanisms of the subject matter described herein are capableof being distributed as a program product in a variety of forms, andthat an illustrative embodiment of the subject matter described hereinapplies regardless of the particular type of signal bearing medium usedto actually carry out the distribution. Examples of a signal bearingmedium include, but are not limited to, the following: a recordable typemedium such as a floppy disk, a hard disk drive, a Compact Disc (CD), aDigital Video Disk (DVD), a digital tape, a computer memory, etc.; and atransmission type medium such as a digital and/or an analogcommunication medium (e.g., a fiber optic cable, a waveguide, a wiredcommunications link, a wireless communication link, etc.).

Those skilled in the art will recognize that it is common within the artto describe devices and/or processes in the fashion set forth herein,and thereafter use engineering practices to integrate such describeddevices and/or processes into data processing systems. That is, at leasta portion of the devices and/or processes described herein may beintegrated into a data processing system via a reasonable amount ofexperimentation. Those having skill in the art will recognize that atypical data processing system generally includes one or more systemunits housing, a video display device, a memory such as volatile andnon-volatile memory, processors such as microprocessors and digitalsignal processors, computational entities such as operating systems,drivers, graphical user interfaces, and applications programs, one ormore interaction devices, such as a touch pad or screen, and/or controlsystems including feedback loops and control motors (e.g., feedback forsensing position and/or velocity; control motors for moving and/oradjusting components and/or quantities). A typical data processingsystem may be implemented utilizing any suitable commercially availablecomponents, such as those typically found in datacomputing/communication and/or network computing/communication systems.

The herein described subject matter sometimes illustrates differentcomponents contained within, or connected with, different othercomponents. It is to be understood that such depicted architectures aremerely exemplary, and that in fact many other architectures may beimplemented which achieve the same functionality. In a conceptual sense,any arrangement of components to achieve the same functionality iseffectively “associated” such that the desired functionality isachieved. Hence, any two components herein combined to achieve aparticular functionality may be seen as “associated with” each othersuch that the desired functionality is achieved, irrespective ofarchitectures or intermedial components. Likewise, any two components soassociated may also be viewed as being “operably connected”, or“operably coupled”, to each other to achieve the desired functionality,and any two components capable of being so associated may also be viewedas being “operably couplable”, to each other to achieve the desiredfunctionality. Specific examples of operably couplable include but arenot limited to physically mateable and/or physically interactingcomponents and/or wirelessly interactable and/or wirelessly interactingcomponents and/or logically interacting and/or logically interactablecomponents.

With respect to the use of substantially any plural and/or singularterms herein, those having skill in the art may translate from theplural to the singular and/or from the singular to the plural as isappropriate to the context and/or application. The varioussingular/plural permutations may be expressly set forth herein for sakeof clarity.

It will be understood by those within the art that, in general, termsused herein, and especially in the appended claims (e.g., bodies of theappended claims) are generally intended as “open” terms (e.g., the term“including” should be interpreted as “including but not limited to,” theterm “having” should be interpreted as “having at least,” the term“includes” should be interpreted as “includes but is not limited to,”etc.). It will be further understood by those within the art that if aspecific number of an introduced claim recitation is intended, such anintent will be explicitly recited in the claim, and in the absence ofsuch recitation no such intent is present. For example, as an aid tounderstanding, the following appended claims may contain usage of theintroductory phrases “at least one” and “one or more” to introduce claimrecitations. However, the use of such phrases should not be construed toimply that the introduction of a claim recitation by the indefinitearticles “a” or “an” limits any particular claim containing suchintroduced claim recitation to inventions containing only one suchrecitation, even when the same claim includes the introductory phrases“one or more” or “at least one” and indefinite articles such as “a” or“an” (e.g., “a” and/or “an” should typically be interpreted to mean “atleast one” or “one or more”); the same holds true for the use ofdefinite articles used to introduce claim recitations. In addition, evenif a specific number of an introduced claim recitation is explicitlyrecited, those skilled in the art will recognize that such recitationshould typically be interpreted to mean at least the recited number(e.g., the bare recitation of “two recitations,” without othermodifiers, typically means at least two recitations, or two or morerecitations). Furthermore, in those instances where a conventionanalogous to “at least one of A, B, and C, etc.” is used, in generalsuch a construction is intended in the sense one having skill in the artwould understand the convention (e.g., “a system having at least one ofA, B, and C” would include but not be limited to systems that have Aalone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). In those instances where aconvention analogous to “at least one of A, B, or C, etc.” is used, ingeneral such a construction is intended in the sense one having skill inthe art would understand the convention (e.g., “a system having at leastone of A, B, or C” would include but not be limited to systems that haveA alone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). It will be furtherunderstood by those within the art that virtually any disjunctive wordand/or phrase presenting two or more alternative terms, whether in thedescription, claims, or drawings, should be understood to contemplatethe possibilities of including one of the terms, either of the terms, orboth terms. For example, the phrase “A or B” will be understood toinclude the possibilities of “A” or “B” or “A and B.”

1. A method for a computing device to create a location discoveryinfrastructure (LDI) for estimating locations of one or more of aplurality of communication nodes in a terrain, comprising: grouping, bythe computing device, the one or more of the plurality of communicationnodes into a plurality of communication node groups, such thatcommunication nodes in individual communication node groups are inproximity and have approximately no obstacles in between; selecting, bythe computing device, a representative communication node from theindividual communication node groups, the selected representativecommunication nodes of the plurality of communication node groupsforming a dominating set of communication nodes; and determining, by thecomputing device, a number and locations of beacon nodes to be placed inthe terrain, based at least in part on individual representativecommunication nodes receiving signals from at least three beacon nodes;wherein S={S_(i)},i=1, . . . , N_(s) denote the plurality ofcommunication nodes to be divided into a plurality of communication nodegroups; and wherein the selecting comprises formulating and solving anobjective function (OF) as follows:${OF}\text{:}\mspace{14mu}\min{\sum\limits_{i = 1}^{N_{s}}\; d_{i}}$${{{such}\mspace{14mu}{that}\text{:}\mspace{14mu}{\sum\limits_{j = 1}^{N_{s}}\;{d_{j}E_{ij}}}} \geq 1},\mspace{14mu}{i = 1},\ldots\mspace{14mu},N_{S},{E_{ij} = \left\{ {\begin{matrix}{1,} & \begin{matrix}{{if}\mspace{14mu}{communication}\mspace{20mu}{nodes}\mspace{14mu}{Si}\mspace{14mu}{and}\mspace{14mu}{Sj}\mspace{14mu}{receive}\mspace{14mu}{signals}} \\{{from}\mspace{14mu}{same}\mspace{14mu}{set}\mspace{14mu}{of}\mspace{14mu}{three}\mspace{14mu}{or}\mspace{14mu}{more}\mspace{14mu}{beacon}\mspace{14mu}{nodes}}\end{matrix} \\{0,} & {otherwise}\end{matrix},{{{and}d_{i}} = \left\{ \begin{matrix}{1,} & {{if}\mspace{14mu}{communication}\mspace{20mu}{nodes}\mspace{14mu}{Si}\mspace{14mu}{is}\mspace{14mu}{selected}\mspace{14mu}{in}\mspace{14mu}{the}\mspace{14mu}{dominating}\mspace{14mu}{set}} \\{0,} & {{otherwise}.}\end{matrix} \right.}} \right.}$
 2. The method of claim 1, wherein thedetermining is further based at least in part on reducing a likelihoodof location discovery error for discovering locations of one or more ofthe plurality of communication nodes using the placed beacon nodes; andwherein the method further comprises determining, by the computingdevice, a schedule for the placed beacon nodes to be active, based atleast in part on a waiting time of individual communication nodes toreceive location determination signals from one or more beacon nodes. 3.The method of claim 1, wherein the determining is further based at leastin part on individual representative communication nodes receivingsignals from at least four of the placed beacon nodes.
 4. The method ofclaim 1, further comprising: identifying, for individual pairs of theplaced beacon nodes, a likelihood of interference or collision at one ormore communication nodes when two beacon nodes of a pair of beacon nodessimultaneously transmits location estimation signals.
 5. The method ofclaim 4, wherein the identifying further comprises: formulating a graphcoloring problem, wherein individual vertices of a graph represent acorresponding beacon node, wherein an edge between two vertices of thegraph represent a condition associated with the two associated beaconnodes transmitting signals simultaneously, and wherein the condition isbased at least in part of identifying the likelihood of interference orcollision for the two beacon nodes; and solving the graph coloringproblem to determine the plurality of beacon node groups.
 6. The methodof claim 1, wherein the determining the schedule comprises: determiningthe schedule for the placed beacon nodes to be active, based at least inpart on reducing the waiting time of individual communication nodes toreceive location determination signals from one or more beacon nodes. 7.The method of claim 1, wherein the determining the number and locationscomprises: determining a minimum number of beacon nodes required to beplaced in the terrain.
 8. The method of claim 1, wherein the determiningthe number and locations comprises: determining the number and locationsof beacon nodes to be placed in a terrain, based at least in part onreducing a likelihood of location discovery error for discoveringlocations of the representative communication nodes using the placedbeacon nodes.
 9. The method of claim 8, wherein determining the numberand locations further comprises: determining the likelihood of locationdiscovery error from a location discovery model that models correlationbetween a number of neighboring nodes of a communication node and alikelihood of location discovery error of the communication node.
 10. Anapparatus for creating a location discovery infrastructure (LDI) forestimating locations of one or more of a plurality of communicationnodes in a terrain, comprising: a memory that stores a plurality ofprogramming instructions; and a processor coupled to the memory andconfigured to execute the plurality of programming instructions toenable the apparatus, in response to execution of the plurality ofprogramming instructions, to: group one or more of the plurality ofcommunication nodes into a plurality of communication node groups, suchthat communication nodes in individual communication node groups are inproximity and have approximately no obstacles in between; select arepresentative communication node from the individual communication nodegroups, the selected representative communication nodes of the pluralityof communication node groups forming a dominating set of communicationnodes; and determine a number and locations of beacon nodes to be placedin the terrain, based at least in part on individual representativecommunication nodes receiving signals from at least three beacon nodes;wherein S={S_(i)},i=1, . . . , N_(s) denote the plurality ofcommunication nodes to be divided into a plurality of communication nodegroups; and wherein selecting comprises solving an objective function(OF) as follows:${OF}\text{:}\mspace{14mu}\min{\sum\limits_{i = 1}^{N_{s}}\; d_{i}}$${{{such}\mspace{14mu}{that}\text{:}\mspace{14mu}{\sum\limits_{j = 1}^{N_{s}}\;{d_{j}E_{ij}}}} \geq 1},\mspace{14mu}{i = 1},\ldots\mspace{14mu},N_{S},{E_{ij} = \left\{ {\begin{matrix}{1,} & \begin{matrix}{{if}\mspace{14mu}{communication}\mspace{20mu}{nodes}\mspace{14mu}{Si}\mspace{14mu}{and}\mspace{14mu}{Sj}\mspace{14mu}{receive}\mspace{14mu}{signals}} \\{{from}\mspace{14mu}{same}\mspace{14mu}{set}\mspace{14mu}{of}\mspace{14mu}{three}\mspace{14mu}{or}\mspace{14mu}{more}\mspace{14mu}{beacon}\mspace{14mu}{nodes}}\end{matrix} \\{0,} & {otherwise}\end{matrix};{{{and}d_{i}} = \left\{ \begin{matrix}{1,} & {{if}\mspace{14mu}{communication}\mspace{20mu}{nodes}\mspace{14mu}{Si}\mspace{14mu}{is}\mspace{14mu}{selected}\mspace{14mu}{in}\mspace{14mu}{the}\mspace{14mu}{dominating}\mspace{14mu}{set}} \\{0,} & {{otherwise}.}\end{matrix} \right.}} \right.}$
 11. The apparatus of claim 10, whereinin response to execution of the programming instructions, the apparatusis further enabled to: determine the number and locations of beaconnodes to be placed in the terrain, based at least in part on reducing alikelihood of location discovery error for discovering locations of oneor more of the plurality of communication nodes using the placed beaconnodes; and determine a schedule for the placed beacon nodes to beactive, based at least in part on a waiting time of individualcommunication nodes to receive location determination signals from oneor more beacon nodes.